I'm working on TA cloning for the first time and I need to turn some existing plasmid backbones into T-vectors. I'm using a Klenow fragment (3' -> 5' exo-) to dA-tail my inserts. Is there any reason why I couldn't use this Klenow fragment to dT-tail my vectors with a similar reaction set up? Thanks in advance for any insight!
Personally I would use Taq DNA polymerase to A overhang my inserts, especially if you are PCR amplifying the insert. It leaves an A overhang on the end of all amplicons natively. All you have to do here is add some Taq at the end of the reaction and incubate for 15 min at 72 $^\circ$C, it'll work in pretty much any PCR buffer and many restriction enzyme buffers too.
Generally it is the vectors that have a T overhang, but this is a legacy of when Taq was pretty much the only polymerase around, so there is no reason you can't have an insert with a T and the vector with the A.
As a general note, often these days it is easier (and cheaper (when you factor in the cost of your time etc) + quicker) to buy a suitable TA cloning vector pre-made, they are highly reliable and should work every time, which is not necessarily something you can say about home-made ones.
Another option, if it fits your workflow, is to add restriction sites to your primers and amplify the insert by PCR. You can then digest the amplified insert and vector with the restriction enzyme of choice and get directional cloning (where your insert always goes in in the same orientation, particularly useful for protein expression). This also allows you to do things like insert short tags (e.g. FLAG, HA, V5), make frame adjustments and even modify the ends of the sequence.